arXiv subjects
Publications and source records attributed to Mohsen Asgharzadeh.
We compute support of formal cohomology modules in a serial of non-trivial cases. Applications are given. For example, we compute injective dimension of certain local cohomology modules in terms of dimension of their's support.
We deal with the rigidity conjecture of symbolic powers over regular rings. This was asked by Huneke. Along with our investigation, we confirm a conjecture [7, Conjecture 3.8].
In this note we compute length, support and dimension of syzygy modules of certain modules. This partially answers questions asked by Huneke et al.
We compute some numerical invariants of local cohomology of the ring of invariants by a finite group, mainly in the modular case. Also, we present some applications. In particular, we study Cohen-Macaulay property of modular invariants from the viewpoints of depth, Serre's condition and the relevant generalizations (e.g., the Buchsbaum property, etc). The situation in the local case is different from the global case.
We deal with some questions posted by Matsumura and Watanabe about the Rees and Dilworth number, and their higher-dimensional versions.
We investigate homological properties of perfect algebras of prime characteristic. The principle is as follows: perfect algebras resolve the singularities. For example, we show any module over the ring of absolute integral closure has finite flat dimension. Under some mild conditions, we show any module over that ring has finite projective dimension. We compute weak dimension and global dimension of perfect rings in a series of nontrivial cases. Some interesting applications are given. In particular, we answer some questions asked by Shimomoto.
Let $k$ be any field. Let $R=k[[X_1,\ldots,X_d]]$ and $\frak p$ a $1$-dimensional prime ideal. In this note we present $d-1$ elements such as $f_1,\ldots,f_{d-1}$ in $\frak p$ such that $\mathfrak{p}=\text{rad}(f_1,\ldots,f_{d-1})$.
We identify families of commutative rings that can be written as a direct limit of a directed system of noetherian regular rings and investigate the homological properties of such rings.
In this short note, we deal with a problem posted by Huneke on a numerical invariant attached to symbolic powers of homogenous ideals.
We prove the (LC) conjecture of Hochster and Huneke in some non-trivial cases. This has several applications. Recently, Brenner and Caminata answered a numerical evidence due to Dao and Smirnov on the shape of generalized Hilbert-Kunz functions of smooth curves. As applications, we first reprove this by a short argument. Then we give a proof of second numerical evidence predicted by Dao and Smirnov on the shape of generalized Hilbert-Kunz functions of nodal curves. Thirdly, we answer a question posted by Vraciu on the (LC) property of a proposed ring. Inspiring with the (LC) property, we present a connection to the stability theory. This leads us to investigate the stability and the strong semistability of the sheaf of relations on $\{x^2,y^2,z^2\}$ over the Klein's quartic curve. This answers questions of Brenner. After presenting a connection from (LC) to the $F$-threshold, we answer a question posted by Huneke et al. Additional applications and examples are given.
In this note we first give a new bound on $e_{HK}(\sim)$ the Hilbert-Kunz multiplicity of invariant rings, by the help of the Noether's bound. Then, we simplify, extend and present applications of the reciprocity formulae due to L. Smith. He proved the formula over polynomial rings and his result is tight in the following sense: Over complete intersection rings with isolated singularity we show that the reciprocity formulae "$e_{HK}(R/I)+ e_{HK}(R/J) = e_{HK}(R/\underline{f})$" is equivalent with $\pd(I)<\infty$ when $I$ is an $\fm$-primary unmixed ideal linked to $J$ along with a regular sequence $\underline{f}$.
Let k be a field and R a pure subring of the infinite-dimensional polynomial ring k[X1;...]. If R is generated by monomials, then we show that the equality of height and grade holds for all ideals of R. Also, we show R satisfies the weak Bourbaki unmixed property. As an application, we give the Cohen-Macaulay property of the invariant ring of the action of a linearly reductive group acting by k-automorphism on k[X1;...]. This provides several examples of non-Noetherian Cohen-Macaulay rings (e.g. Veronese, determinantal and Grassmanian rings).
We consider the following question: Is Gorenstein homology a X-pure homology, in the sense defined by Warfield, for a class X of modules? Let GP denote the class of Gorenstein projective modules. We prove that over a commutative Noetherian ring R of finite Krull dimension, Gorenstein homology is a GP-pure homology if and only if R is virtually Gorenstein.
Let $A$ be a direct limit of a direct system of Cohen-Macaulay rings. In this paper, we describe the Cohen-Macaulay property of $A$. Our results indicate that $A$ is not necessarily Cohen-Macaulay. We show $A$ is Cohen-Macaulay under various assumptions. As an application, we study Cohen-Macaulayness of non-affine normal semigroup rings.
In this paper, we study the Cohen-Macaulayness of non-affine normal semigroups in $\mathbb{Z}^n$. We do this by establishing the following four statements each of independent interest: 1) a Lazard type result on $I$-supported elements of $\prod_{\mathbb{N}}\mathbb{Q}_{\geq0}$ for an index set $I\subset\mathbb{N}$; 2) a criterion of regularity of sequences of elements of the ring via projective dimension; 3) a direct limit of polynomial rings with toric maps; 4) any direct summand of rings of the third item is Cohen-Macaulay. To illustrate the idea, we give many examples.
For a Noetherian local domain $R$ let $R^+$ be the absolute integral closure of $R$ and let $R_{\infty}$ be the perfect closure of $R$, when $R$ has prime characteristic. In this paper we investigate the projective dimension of residue rings of certain ideals of $R^+$ and $R_{\infty}$. In particular, we show that any prime ideal of $R_{\infty}$ has a bounded free resolution of countably generated free $R_{\infty}$-modules. Also, we show that the analogue of this result is true for the maximal ideals of $R^+$, when $R$ has residue prime characteristic. We compute global dimensions of $R^+$ and $R_{\infty}$ in some cases. Some applications of these results are given.
The present paper deals with various aspects of the notion of almost Cohen-Macaulay property, which was introduced and studied by Roberts, Singh and Srinivas. We employ the definition of almost zero modules as defined by a value map, which is different from the version of Gabber-Ramero. We prove that, if the local cohomology modules of an algebra $T$ of certain type over a local Noetherian ring are almost zero, $T$ maps to a big Cohen-Macaulay algebra. Then we study how the almost Cohen-Macaulay property behaves under almost faithfully flat extension. As a consequence, we study the structure of $F$-coherent rings of positive characteristic in terms of almost regularity.
This paper deals with the notion of grade of ideals with respect to torsion theories defined via some homological tools such as Ext-modules, Koszul cohomology modules, Čech and local cohomology modules over commutative rings which are not necessarily Noetherian. We also compare these approaches of grade.